We've had a video attribute on recordings for a while now and we've added support for data tracks which often contain videos, so I think it's safe to say video recordings aren't going anywhere any time soon. With that in mind, it'd be really nice to be able to link an audio recording to the music videos for it.
Right now, they can be linked to the same work, but there's nothing saying they're the same version of the song - even if they're the same artist, one could be a re-recording or something, so finding a list of music videos for a song is pretty hard to do. That also means that if you add a streaming URL to the video, you can't actually get to it from the audio recording, even though being able to do that would be useful.
For link phrases, I would probably suggest "music videos", "music video for" and "has music video".
For the description, maybe something like "This links a music video to the corresponding audio recording."
I don't think it should require the audio to be absolutely identical, because many videos include things like extra special effect sounds. It would make it really hard to be sure that the relationship is correct and I can't imagine that it would be very useful to someone trying to find music videos for a recording anyway. I'm not sure whether I would put that in the description or in the guidelines though.
https://beta.musicbrainz.org/search?query=comment%3A%28pv+%22music+video%22+mv%29+recording%3A%22music+video%22&type=recording&limit=25&method=advanced is probably almost entirely things which could use this relationship type, and between data tracks, bonus DVDs and YouTube, I'm sure I could find hundreds (if not thousands) more examples if I had to, so I think there's a clear use case for this and plenty of data which could use it.